Rank 3 incidence structures admitting dual-linear, linear diagram (Q1062271)

The following result is proved: Let \(\Gamma\) be a finite rank 3 Buekenhout incidence structure admitting diagram \(\circ\overset{^{L^*}}\diagrbar\circ\overset{^L}\diagrbar\circ\). Then for some generalized projective geometry \(\pi\) and some integer \(i\), \(\Gamma\) is isomorphic to the rank 3 Buekenhout incidence structure having all \((i-2)\)-, \((i-1)\)-, and \(i\)-dimensional subspaces of \(\pi\) as varieties, and comparability as the incidence relation. An analogous result is given for a \(\Gamma\) admitting diagram \(\circ\overset{^{c^*}}\diagrbar\circ\overset{^c}\diagrbar\circ\) for some set X.